Kepler Object of Interest Network I. First results combining ground and space-based observations of Kepler systems with transit timing variations
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Kepler Object of Interest Network I. First results combining ground and space-based observations of Kepler systems with transit timing variations von Essen, C., Ofir, A., Dreizler, S., Agol, E., Freudenthal, J., Hernandez, J., Wedemeyer, S., Parkash, V., Deeg, H. J., Hoyer, S., Morris, B. M., Becker, A. C., Sun, L., Gu, S. H., Herrero, E., Tal-Or, L., Poppenhaeger, K., Mallonn, M., Albrecht, S., ... Wang, X. (2018). Kepler Object of Interest Network I. First results combining ground and space-based observations of Kepler systems with transit timing variations. Astronomy & Astrophysics. https://doi.org/10.1051/0004-6361/201732483 Published in: Astronomy & Astrophysics Document Version: Peer reviewed version Queen's University Belfast - Research Portal: Link to publication record in Queen's University Belfast Research Portal Publisher rights © ESO 2018. This work is made available online in accordance with the publisher’s policies. Please refer to any applicable terms of use of the publisher. General rights Copyright for the publications made accessible via the Queen's University Belfast Research Portal is retained by the author(s) and / or other copyright owners and it is a condition of accessing these publications that users recognise and abide by the legal requirements associated with these rights. Take down policy The Research Portal is Queen's institutional repository that provides access to Queen's research output. Every effort has been made to ensure that content in the Research Portal does not infringe any person's rights, or applicable UK laws. If you discover content in the Research Portal that you believe breaches copyright or violates any law, please contact openaccess@qub.ac.uk. Download date:26. Jan. 2022
Astronomy & Astrophysics manuscript no. KOINet c ESO 2018
January 22, 2018
Kepler Object of Interest Network
I. First results combining ground and space-based observations of Kepler
systems with transit timing variations
C. von Essen1,2 , A. Ofir2,3 , S. Dreizler2 , E. Agol4,5,6,7 , J. Freudenthal2 , J. Hernández8 , S. Wedemeyer9,10 , V. Parkash11 ,
H. J. Deeg12,13 , S. Hoyer12,13,14 , B. M. Morris4 , A. C. Becker4 , L. Sun15 , S. H. Gu15 , E. Herrero16 , L. Tal-Or2,18 , K.
Poppenhaeger19 , M. Mallonn20 , S. Albrecht1 , S. Khalafinejad21 , P. Boumis22 , C. Delgado-Correal23 , D. C. Fabrycky24 ,
R. Janulis25 , S. Lalitha26 , A. Liakos22 , Š. Mikolaitis25 , M. L. Moyano D’Angelo27 , E. Sokov28,29 , E. Pakštienė25 , A.
arXiv:1801.06191v1 [astro-ph.EP] 18 Jan 2018
Popov30 , V. Krushinsky30 , I. Ribas16 , M. M. Rodrı́guez S.8 , S. Rusov28 , I. Sokova28 , G. Tautvaišienė25 , X. Wang15
(Affiliations can be found after the references)
Received 18/12/2017; accepted
ABSTRACT
During its four years of photometric observations, the Kepler space telescope detected thousands of exoplanets and exoplanet candidates. One
of Kepler’s greatest heritages has been the confirmation and characterization of hundreds of multi-planet systems via Transit Timing Variations
(TTVs). However, there are many interesting candidate systems displaying TTVs on such long time scales that the existing Kepler observations are
of insufficient length to confirm and characterize them by means of this technique. To continue with Kepler’s unique work we have organized the
“Kepler Object of Interest Network” (KOINet), a multi-site network formed by several telescopes spread among America, Europe and Asia. The
goals of KOINet are to complete the TTV curves of systems where Kepler did not cover the interaction timescales well, to dynamically prove that
some candidates are true planets (or not), to dynamically measure the masses and bulk densities of some planets, to find evidence for non-transiting
planets in some of the systems, to extend Kepler’s baseline adding new data with the main purpose of improving current models of TTVs, and to
build a platform that can observe almost anywhere on the Northern hemisphere, at almost any time. KOINet has been operational since March,
2014. Here we show some promising first results obtained from analyzing seven primary transits of KOI-0410.01, KOI-0525.01, KOI-0760.01,
and KOI-0902.01 in addition to Kepler data, acquired during the first and second observing seasons of KOINet. While carefully choosing the
targets we set demanding constraints about timing precision (at least 1 minute) and photometric precision (as good as 1 part per thousand) that
were achieved by means of our observing strategies and data analysis techniques. For KOI-0410.01, new transit data revealed a turn-over of its
TTVs. We carried out an in-depth study of the system, that is identified in the NASA’s Data Validation Report as false positive. Among others,
we investigated a gravitationally-bound hierarchical triple star system, and a planet-star system. While the simultaneous transit fitting of ground
and space-based data allowed for a planet solution, we could not fully reject the three-star scenario. New data, already scheduled in the upcoming
2018 observing season, will set tighter constraints on the nature of the system.
Key words. stars: planetary systems – methods: observational
1. Introduction Wolszczan 1994; Laughlin & Chambers 2001; Rivera et al.
2010; Holman et al. 2010; Lissauer et al. 2011a; Becker et al.
Transit observations provide a wealth of information about 2015; Gillon et al. 2016). Some examples of ground-based
alien worlds. Beside the detection and characterization of ex- transit timing variation (TTV) studies are WASP-10b
oplanets (e.g. Seager 2010), once an exoplanet is detected by (Maciejewski et al. 2011), WASP-5b (Fukui et al. 2011), WASP-
its transits the variations of the observed mid-transit times 12b (Maciejewski et al. 2013), and WASP-43b (Jiang et al.
can be used to characterize the dynamical state of the system 2016). Accompanying the observational growth, theoretical and
(Holman & Murray 2005; Agol et al. 2005). The timings of a numerical models were developed to reproduce the timing shifts
transiting planet can sometimes be used to derive constraints on and represent the most probable orbital configurations (e.g.,
the planetary physical and orbital parameters in the case of mul- Agol et al. 2005; Nesvorný & Morbidelli 2008; Lithwick et al.
tiple transiting planets (Holman et al. 2010), to set constraints 2012; Deck et al. 2014). There is no doubt about the detection
on the masses of the perturbing bodies (Ofir et al. 2014), and to power of the TTV method: given the mass of the host star, ana-
characterize the mass and orbit of a non-transiting planet, with lyzing photometric observations we can sometimes retrieve the
masses potentially as low as an Earth mass (Agol et al. 2005; orbital and physical properties of complete planetary systems
Nesvorný et al. 2013; Barros et al. 2013; Kipping et al. 2014; (Carter et al. 2012). However, the method requires sufficiently
Jontof-Hutter et al. 2015). For faint stars, this is extremely chal- long baseline, precise photometry and good phase coverage.
lenging to achieve by means of other techniques.
In the past three decades, non-Keplerian motions of ex- From ground-based studies, which have focused on TTVs
oplanets have been regularly studied from the ground and of hot Jupiters, there have already been some discrepant re-
space (Rasio et al. 1992; Malhotra et al. 1992; Peale 1993; sults (see e.g. Qatar-1, von Essen et al. 2013; Mislis et al. 2015;
Collins et al. 2017), especially when small-sized telescopes are
Send offprint requests to: cessen@phys.au.dk involved and TTVs of low amplitude are being measured
1C. von Essen et al. (2017): KOINet: study of exoplanet systems via TTVs
(von Essen et al. 2016). Also, many follow-up campaigns of hot Section 4 makes special emphasis to the fitting strategy of both
Jupiters could not significantly observe TTVs from the ground ground and space-based data. In Section 5 we show KOINet’s
(see e.g. Steffen & Agol 2005; Fukui et al. 2016; Petrucci et al. achieved milestones, and we finish with Section 6, where we
2015; Raetz et al. 2015; Mallonn et al. 2015, for TrES-1, HAT- present our conclusions and a brief description of the future ob-
P-14b, WASP-28b, WASP-14b, and HAT-P-12b respectively). serving seasons of KOINet.
However, hot Jupiters tend to be isolated from companion plan-
ets (Steffen et al. 2012b) so it does not come as a surprise that
these studies have not resulted in convincing signals. It was
with the advent of space-based observatories that a new era 2. Kepler Object of Interest Network
in the TTV quest started. In March 2009, NASA launched the
Kepler space telescope (Borucki et al. 2010; Koch et al. 2010). 2.1. Rationale
The main goal of the mission was to detect Earth-sized planets
in the so-called habitable zone, orbiting around stars similar to KOINet’s unique characteristic is the use of already existing
our Sun. The wide field of view allowed simultaneous and con- telescopes, coordinated to work together towards a common
tinuous monitoring of many thousands of stars for about four goal. The data collected by the network will provide three ma-
years. Surprisingly, Kepler showed a bounty of planetary sys- jor contributions to the understanding of the exoplanet popu-
tems with a much more compact configuration than our Solar lation. First, deriving planetary masses from transit timing ob-
System (Lissauer et al. 2014). About 20% of the known plane- servations for more planets will populate the mass-radius dia-
tary systems present either more than one planet or more than gram. The distribution of planetary radii at a given planetary
one star (Fabrycky et al. 2014). Particularly, most multiple sys- mass is surprisingly wide, revealing a large spread in internal
tems are formed by at least two planets and about one third of compositions (see e.g. Mordasini et al. 2012). New mass and
these appear to be close to mean motion resonant orbits (see radius determinations will provide new constraints for planet
Lissauer et al. 2011b). Thus, the long-term and highly precise structure models. Furthermore, longer transit monitoring will
observations provided by Kepler have been the most success- set tighter constraints for the existence of non-transiting plan-
ful data source used to confirm and characterize planetary sys- ets (Barros et al. 2014), providing a broader and deeper view of
tems via TTVs. Preceding a very long list, the first example of the architecture of planetary systems. Finally, a larger sample of
outstanding TTV discoveries is Kepler-9 (Holman et al. 2010). well-constrained physical parameters of planets and planetary
Since then, several other planetary systems were confirmed, systems will provide better constraints for their formation and
detected or even characterized by means of TTV studies (see evolution (Lissauer et al. 2011b; Fang & Margot 2012).
e.g. Hadden & Lithwick 2014; Nesvorný et al. 2014). Classic KOINet is initially focusing its instrumental resources on 60
examples are Kepler-11 (Lissauer et al. 2011a), Kepler-18 KOIs that require additional data to complete a proper charac-
(Cochran et al. 2011), Kepler-19 (Ballard et al. 2011), Kepler-23 terization or validation by means of the TTV technique. Basic
and Kepler-24 (Ford et al. 2012a), Kepler-25 to Kepler-28 information on the selected KOIs can be seen in the left part
(Steffen et al. 2012a), Kepler-29 to Kepler-32 (Fabrycky et al. of Table 2. The KOI target list was built up based on the work
2012), and Kepler-36 (Carter et al. 2012). The list goes of Ford et al. (2012b), Mazeh et al. (2013), Xie (2013, 2014),
up to Kepler-87 (Ofir et al. 2014) and continues with K2, Nesvorný et al. (2013), Ofir et al. (2014), and Holczer et al.
Kepler’s second chance at collecting data that will allow (2016). The 60 KOIs were drawn from four groups, depending
us to investigate planetary systems by means of TTVs (see on the scientific insights that further observations were expected
e.g. Becker et al. 2015; Nespral et al. 2016; Jontof-Hutter et al. to provide.
2016; Hadden & Lithwick 2017). Mazeh et al. (2013) analyzed For a pair of planets, an anti-correlation in the TTV signal
the first twelve quarters of Kepler photometry and derived the is expected to occur. This is the product of conservation of en-
transit timings of 1960 Kepler objects of interest (KOIs). An up- ergy and angular momentum and is stronger when the plane-
dated analysis of Kepler TTVs using the full long-cadence data tary pair is near mean-motion resonance (see e.g. Holman et al.
set can be found under Holczer et al. (2016). The authors found 2010; Carter et al. 2012; Lithwick et al. 2012). The systems that
that 130 KOIs presented significant TTVs, either because their present polynomial-shaped TTVs and show anti-correlated TTV
mid-transit times had a large scatter, showed a periodic mod- signals are given the highest priority, independent of their sta-
ulation, or presented a parabola-like trend. Although ∼80 KOIs tus as valid planet candidates. In these cases, any additional
showed a clear sinusoidal variation, for other several systems the data points in their parabolic-shaped TTVs can reveal a turn-
periodic signal was too long in comparison with the time span of over point, allowing a more accurate determination of plane-
Kepler data to cover one full TTV cycle. As a consequence, no tary masses. Further data will allow the analysis of the sys-
proper dynamical characterization could be carried out. tem’s dynamical characteristics. The systems that present anti-
To overcome this drawback and expand upon Kepler’s her- correlation and a sinusoidal variation, but are poorly sampled,
itage, in the framework of a large collaboration we organized the have second priority (such as KOI-0880.01/02, a detailed analy-
Kepler Object of Interest Network1 (KOINet). The main purpose sis of the system is in prep). In this case, more data points will
of KOINet is the dynamical characterization of selected KOIs allow us to improve the dynamical analysis of these systems.
showing TTVs. To date, the network is comprised of numer- Under third priority fall the KOIs with very long TTV period-
ous telescopes and is continuously evolving. KOINet’s first light icity. Additional data might shed some light into the constitu-
took place in March, 2014. Here we show representative data tion of these systems (for example, KOI-0525.01, Section 5.3).
obtained during our first and second observing seasons that will Finally, the lowest priority is given to those systems that have
highlight the need for KOINet. Section 2 shows the basic work- been already characterized, and the systems showing only one
ing structure of KOINet and the scientific milestones, Section 3 TTV signal (e.g., KOI-0410.01, Section 5.5). In the latter, under
describes the observing strategy and the data reduction process. specific conditions the perturber’s mass and orbital period can be
constrained, confirming its planetary nature or ruling it out (e.g.,
1
koinet.astro.physik.uni-goettingen.de Nesvorný et al. 2013, 2014).
2C. von Essen et al. (2017): KOINet: study of exoplanet systems via TTVs
2.2. Observing time Considering these two fundamental limitations, to maximize the
use of KOINet data and boost transit detection we have included
During the first two observing seasons (April-September, 2014 the KOIs whose transit depth are larger than one part per thou-
and 2015) an approximate total of 600 hours were collected for sand (ppt) and which Kepler timing variability (this is, the vari-
our project, divided between 16 telescopes and 139 observing ability comprised within Kepler time span) is larger than two
events. Rather than following up all of the KOIs, we focused minutes (see Figure 2). Below these limits, the photometric pre-
on the most interesting ones from a dynamical point of view. cision (and thus, the derived timing precision), and especially the
Although here we present a general overview of the data col- impact of correlated noise on photometric data (Carter & Winn
lected by KOINet and its performance, we will focus in the anal- 2009) would play a fundamental role in the detection of tran-
ysis of individual KOIs in upcoming publications. sit events. Next, we describe the primary characteristics of the
telescopes involved in this work.
2.3. Basic characteristics of KOINet’s telescopes
Kepler planets and planet candidates showing TTVs generally
present two major disadvantages for ground-based follow-up
observations. On one hand, their host stars are relatively faint 16
1000
TTV semi-amplitude (min)
(K p ∼12-16). On the other hand, most of the KOIs reported to
have large amplitude TTVs produce shallow primary transits. To 15
collect photometric data with the necessary precision to detect
shallow transits in an overall good cadence, most of KOINet’s 100 14
Kmag
telescopes have relatively large collecting areas. This allows to
collect data at a frequency of some seconds to a few minutes. 13
Another observational challenge comes with the transit duration. 10
For some of the KOIs the transit duration is longer than the astro- 12
nomical night, especially bearing in mind that the Kepler field is
best observable around the summer season, when the nights are 1 11
intrinsically shorter. In these cases full transit coverage can only 0.1 1.0 3.0 9.0
be obtained combining telescopes well separated in longitude. Transit depth (ppt)
The telescopes included in this collaboration are spread between
America, Europe and Asia, allowing almost 24 hours of contin-
uous coverage. A world map including the telescopes that col- Fig. 2: Colored rectangles show TTV Kepler variability in min-
lected data during 2014 and 2015 can be found in Figure 1 and utes versus transit depth in ppt for the 60 KOIs that are included
Table 1. in KOINet. The squares are color-coded depending on the Kepler
magnitude of the host star. Black circles show all the KOIs pre-
senting TTVs with a Kepler variability larger than 1 minute.
Vertical and horizontal dashed lines indicate the ∼1 ppt and 2
minutes limits for KOINet.
11
10 12
89 13
1 23 5,6,7 14
16 – The Apache Point Observatory, located in New Mexico,
15 United States of America, hosts the Astrophysical Research
4 Consortium 3.5 meter telescope, henceforth ARC 3.5m. The
data were collected using Agile (Mukadam et al. 2011).
Fig. 1: World’s northern hemisphere, showing approximate lo- Concerning the data presented in this work, the ARC 3.5m
cations of the observing sites that acquired data for KOINet dur- observed one transit of KOI-0525.01, our lower-limit KOI
ing the 2014 and 2015 seasons. Numbers correspond to Table 1. for transit depth. Nonetheless, during the first observing sea-
sons we have collected a substantial amount of data that will
be presented in future work.
– The Nordic Optical Telescope (henceforth NOT 2.5m) is
located at the observatory “Roque de los Muchachos”
Table 1: Placement of the observatories that collected data dur- in La Palma, Spain, and belongs to the Nordic Optical
ing the 2014 and 2015 observing seasons. Telescope Scientific Association, governed and funded by
1 Multiple Mirror Telescope Observatory (6.5m), United States of America
Scandinavian countries. In this work we present observations
2 Apache Point Observatory (3.5m), United States of America of KOI-0760.01 and KOI-0410.01.
3 Monitoring Network of Telescopes (1.2m), United States of America – The 2.2m Calar Alto telescope is located in Almerı́a, Spain
4 Observatorio Astronómico Nacional del Llano del Hato (1m), Venezuela
5 Nordic Optical Telescope (2.5m), Spain (henceforth CAHA 2.2m). We observed KOI-0410.01 using
6 Liverpool Telescope (2m), Spain the Calar Alto Faint Object Spectrograph in its photometric
7 IAC80 telescope (0.8m), Instituto de Astrofı́sica de Canarias, Spain
8 Calar Alto Observatory (1.25, 2.2, 3.5 m), Spain mode.
9 Planetary Transit Study Telescope (0.6m), Spain – The IAC80 telescope (henceforth, IAC 0.8m) is located at
10 Joan Oró Telescope - The Montsec Astronomical Observatory (0.8m), Spain
11 Oskar Lühning Telescope - Hamburger Sternwarte (1.2m), Germany
the Observatorio del Teide, in the Canary Islands, Spain. We
12 Bologna Astronomical Observatory (1.52m), Italy observed half a transit of KOI-0902.01 for about 7 hours.
13 Kryoneri Observatory - National Observatory of Athens (1.2m), Greece – The Lijiang 2.4m telescope (henceforth YO 2.4m) is located
14 Wise Observatory - Tel-Aviv University (1m), Israel
15 IUCAA Girawali Observatory (2m), India at the Yunnan Observatories in Kunming, China. In this work
16 Yunnan Observatories (2.4m), China we present observations of KOI-0410.01.
3C. von Essen et al. (2017): KOINet: study of exoplanet systems via TTVs
2.4. Maximizing the use of KOINet’s time of reference stars can limit the precision of photometric data
(Young et al. 1991; Howell 2006). The true constancy of refer-
2.4.1. Assigning telescopes to KOIs ence stars is given by how much they intrinsically vary, subject
In order to effectively distribute the available telescope time and to the precision that a given optical setup can achieve. Analyzing
maximize our chances to detect transit events, three main char- the flux measurements along the 17 quarters of all the stars
acteristics have to be considered: the apparent magnitude of the within a radius of 5 arcmin relative to KOINet’s KOIs, we se-
host star, the available collecting area given by the size of the lected stars that showed a constant flux behavior in time and had
primary mirror, and the amplitude and scatter of Kepler TTVs. a comparable brightness to the given KOI (Howell 2006). In this
With the main goal to connect the KOIs to the most suitable way, we provide to the observer the location of the most pho-
telescopes, we proceed as follows. First, we estimate the ex- tometrically well-behaved reference stars, minimizing the noise
posure time, Et , for each host star and telescope. The latter is budget right from the beginning. Particularly, we have identified
computed to achieve a given signal-to-noise ratio (SNR), so that between 2 to 5 reference stars per field of view, and their loca-
SNR = 1/Tdepth is satisfied. In this case, Tdepth corresponds to tion on sky are provided to the observers through KOINet’s web
the transit depth in percentage, which is taken from the NASA interface.
Exoplanet Archive2 . Besides the desired signal-to-noise ratio,
the calculation of Et is carried out considering parameters such 2.5. Predictions computed from Kepler timings
as the mean seeing of the site, the brightness of the star, the size
of the primary mirror, typical sky brightness of the observato- Using the mid-transit times obtained from Kepler 17 quarters,
ries, the phase of the Moon, and the altitude of the star during we computed TTVs subtracting from them an averaged (con-
the predicted observing windows. Once the exposure times are stant) period, and classified the KOIs depending on the shape of
computed, the derived values are verified and subsequently con- their TTVs. A full description of the fitting process of Kepler
firmed by each telescope leader. transit light curves, the derived values, and their associated er-
Off-transit data have a considerable impact in the determina- rors, can be found in Section 4.1. For now, Figure 3 shows our
tion of the orbital and physical parameters of any transiting sys- four target groups. The simplest case, in which the TTVs follow
tem. In the case of ground-based observations, off-transit data a sinusoidal shape, is shown on the top left panel of the Figure.
are critical to remove systematic effects related to changes in To estimate the predictions for our ground-based follow-up we
airmass, color-dependent extinction, and poor guiding and flat- fitted to Kepler mid-transit times a linear plus a sinusoidal term:
fielding (see e.g., Southworth et al. 2009; von Essen et al. 2016).
Henceforth, to determine the number of data points per transit, T T V(E) = T 0 (E = 0) + PC × E + A × sin[2π(ν E + φ)]. (2)
N, we use the estimated exposure time and the known transit
duration, Tdur , incremented by two hours. This increment ac- In this case, E corresponds to the transit epoch, T0 (E = 0) to a
counts for 1 hour of off-transit data before and after transit be- reference mid-transit time, PC is the orbital (constant) period,
gins and ends, respectively. Then, the number of data points A the semi-amplitude of the TTVs, and ν and φ the frequency
per transit is simply estimated as N = (Tdur + 2 hs)/(Et + ROT). and phase of the TTVs, respectively. The derived predictions are
Here, ROT corresponds to the readout time of charge-coupled shown in Figure 3 in green points, while Kepler data is plotted
devices used to carry out the observations. To compute the tim- in red and the shape of the predictions, including Kepler time,
ing precision, σT , we use a variant of the formalism provided by is shown in continuous black line. Since all Kepler mid-times
Ford & Holman (2007): show some scatter, we also estimated errors in the predictions
taking this noise into consideration. To increase the chance of
PhotP × Tdur transit detection, the magnitudes of the errors in the predictions
σT = , (1) are provided to the observers, along with a warning. The second
N1/2 × Tdepth
TTV scenario is shown in the top right panel of Figure 3. In this
where PhotP is the photometric precision in percentage that a case the available data and the systems themselves allow a more
given telescope can achieve while observing a 14-15 Kp star. refined dynamical analysis of the TTVs by means of n-body sim-
This value was requested to the members of KOINet im- ulations and/or simultaneous transit fitting (see e.g., Agol et al.
mediately after they joined the network. Comparing the esti- 2005; Nesvorný et al. 2013, 2014), from which the predictions
mated timing precision with the semi-amplitude of Kepler TTVs are computed. Due to their complexity, a detailed description of
(ATTVs > 3σT ) yields erroneous results, especially if the TTVs the computation of these TTVs is beyond the scope of this paper,
are intrinsically large. For example, an estimated timing preci- and will be given individually in future publications. The third
sion of one hour satisfies the above condition for a TTV semi- case is shown in the bottom left panel of Figure 3. Here, the
amplitude of 3 hours. However, when ground-based photometry number of available Kepler transits is not sufficient to carry out
is being analyzed, a timing precision of one hour would be equal a dynamical analysis, and the TTVs don’t follow any shape that
to a non-detection. Therefore, to assign a KOI to a telescope could give us a hint of when could the upcoming transits occur.
three aspects are simultaneously considered: the transit depth Thus, to determine the predictions we fit to Kepler mid-times a
(Tdepth > PhotP ), the amplitude of Kepler TTVs (ATTVs > 3σT ), linear trend only (i.e., assuming constant period), and use as er-
and the natural scatter of Kepler TTVs (2σTTVs > σT ). rors for the predictions the semi-amplitude of the TTVs. The last
case exemplifies the need for a ground-based campaign taking
place immediately after Keplers follow-up. This case, displayed
2.4.2. Prescription for optimum reference stars
in the bottom right panel of Figure 3, shows an incomplete cov-
Differential photometry highlights the variability of one star erage of the TTV periodicity. From photometry only we cannot
(the so-called target star) relative to another one (the reference assess if the cause for TTVs is planetary in nature, is gravita-
star) which ideally should not vary in time. Thus, the selection tionally bound to the system (e.g., TTVs following a sinusoidal
shape), or some completely different scenario, like TTVs caused
2
https://exoplanetarchive.ipac.caltech.edu by a blended eclipsing binary (TTVs showing a parabolic shape).
4C. von Essen et al. (2017): KOINet: study of exoplanet systems via TTVs
In this case, we produce two kind of predictions: sine TTVs, the lowest possible value for the annulus. To perform a posterior
from where the predictions are computed as described in Eq. 2, detrending of the photometric data, in addition to Ŝ , the pipeline
and parabolic TTVs: computes the airmass corresponding to the center of the field of
view, the (x,y) centroid positions of all the measured stars, three
T T V(E) = T 0 (E = 0) + PC × E + a × E 2 + b × E + c. (3) sky values originally used to compute the integrated fluxes (one
where a, b, and c are the fitting coefficients of the parabola. per sky ring), and the integrated counts of the master flat and
Although these are the two scenarios most likely to occur, the master dark over the (x,y) values per frame and per aperture.
mid-times could also show a different trend. Therefore, until we The second part of DIP2 OL is python-based. The routine
can disentangle which trend is the one that the system follows, starts by producing N+1 light curves from the N reference fluxes
we provide to the observers both predictions and ask them to ob- previously computed by IRAF, one with the summed flux of all
serve both of them, and extend the observing time as much as the N comparison stars and N versions with all the reference stars
they can. except one. If one of the reference stars is photometrically un-
stable, the residual light curve corresponding to the unweighted
sum of the fluxes of all the reference stars minus this one will
3. Observations and data reduction show up by giving the lowest standard deviation, when com-
pared to the remaining N residuals. Therefore, this star is re-
3.1. Basic observing setup
moved from the sample. The process of selection and rejection is
In order to ensure observations as homogeneous as possible, ob- repeated until the combination of the current available reference
servers are asked to carry them out in a specific way. To be- stars gives the lowest scatter in the photometry. Since a priori
gin with, our observations cover a range of airmass and so are we don’t know if primary transits are actually observed within a
subject to differential extinction effects between the target and given predicted window, residuals are computed by dividing the
comparison stars. To minimize color-dependent systematic ef- differential fluxes by a spline function. The pipeline repeats this
fects observers used intermediate (Cousins R) or narrow-band process through all measured apertures and sky rings, and finds
(gunn r) filters, depending on the brightness of the target stars the combination of reference stars, aperture and sky ring that
and filter availability. The use of R-band filters also reduces minimizes the standard deviation of the differential light curves
light curve variations from starspots and limb-darkening effects, (see e.g., Ofir et al. 2014). Finally, the code outputs the time in
and they circumvent the large telluric contamination around the Julian dates shifted to the center of the exposure, the differen-
I-band. Furthermore, all observers provide regular calibrations tial fluxes, photometric error bars which magnitudes have been
(bias flatfield frames and darks, if needed), and are asked to ob- scaled to match the standard deviation of the residuals, (x,y) cen-
serve with the telescope slightly defocused to minimize the noise troid positions, flat counts that were integrated within the final
in the photometry (Kjeldsen & Frandsen 1992; Southworth et al. aperture around the given centroids, sky fluxes corresponding to
2009). Once the observations are performed, they are collected the chosen sky ring, and seeing and airmass values. These quan-
and reduced in an homogeneous way. tities will be used in a following step to compute the ground-
based detected mid-transit times.
3.2. DIP2 OL
4. Data modelling and fitting strategies
KOINet data are reduced and analyzed by means of the
Differential Photometry Pipelines for Optimum Lightcurves, 4.1. Primary transit fitting of Kepler data
DIP2 OL. The pipeline is divided in two parts. The first one is One of the key ingredients for the success of our ground-based
based in IRAF’s command language. It requires only one refer- TTV follow-up is the prior knowledge, with a good degree of
ence frame to do aperture photometry. The pipeline carries out accuracy, of the orbital and physical parameters of the sys-
normal calibration sequences (bias and dark subtraction and flat- tems. To take full advantage of Kepler data in our work, we re-
field division, depending on availability) using IRAF task ccd- computed the orbital and physical parameters of the 60 KOIs
proc. In the particular case of KOINet data, acquired calibra- that are included in KOINet’s follow-up. A quick view into the
tions are always a set of bias and flatfields, taken either at the Data Validation Reports suggested us that the procedures per-
beginning or end of each observing night. Subject to availability, formed over KOIs without TTVs was not optimum for KOIs
we correct the science frames of a given observing night with showing TTVs. Thus, we did not use the transit parameters re-
their corresponding calibrations only. In general, we do not take ported by the NASA Exoplanet Archive. Rather than computing
dark frames due to short exposures and cooled, temperature sta- time-expensive photo-dynamical solutions over the 60 KOIs (see
ble CCDs. The reduction continues with cosmic rays rejection e.g., Barros et al. 2015), to minimize the impact of the TTVs in
(IRAF’s cosmicrays) and alignment of the science frames (ima- the computation of the transit parameters we fitted two conse-
lign). Afterwards, reference stars within the field are chosen fol- quent transit light curves simultaneously with a Mandel & Agol
lowing specific criteria (for example, that the brightness of the (2002) transit model, making use of their occultquad routine3 .
reference stars have to be similar to the brightness of the target From the transit light curve we can determine the following pa-
star to maximize the signal-to-noise ratio of the differential light rameters: the orbital period, Per, the mid-transit time, T0 , the
curves, Howell 2006) and photometric fluxes and errors are mea- planet-to-star radius ratio, Rp /Rs , the semi-major axis in stel-
sured over the target star and the reference stars as a function of lar radius, a/Rs , and the orbital inclination, i, in degrees. For
10 different aperture radii and 3 different sky rings. The annu- all the KOIs we assumed circular orbits. Furthermore, we as-
lus and the initial width of the sky ring are set by the user, since sumed a quadratic limb-darkening law with fixed limb dark-
they depend on the crowding of the fields. The apertures are non- ening coefficients, u1 and u2 . For the Kepler data we used the
uniformly distributed between 0.5 and 5×Ŝ , with more density limb-darkening values specified in Claret et al. (2013), choos-
between 1 and 2×Ŝ . Here, Ŝ corresponds to the averaged seeing ing as fundamental stellar parameters, effective temperature,
of the images, computed from the full-width at half maximum
3
of all the chosen stars in the field. This, in turn, sets a limit to http://www.astro.washington.edu/users/agol
5C. von Essen et al. (2017): KOINet: study of exoplanet systems via TTVs
30 Kepler data KOI-0250.01 0.6
Predictions
20 0.4
TTVs [days]
10 0.2
TTVs [min]
0 0
-10 -0.2
-20 -0.4
Kepler data KOI-0142.01
-30 -0.6 Predictions
0 500 1000 1500 2000 2500 0 500 1000 1500 2000 2500
Time [days; BJD - 2454833] Time [days; BJD - 2454833]
35
Kepler data KOI-0372.01 Kepler data KOI-0410.01
20 Predictions 30 Predictions
25
10 20
TTVs [min]
TTVs [min]
15
0
10
-10 5
0
-20 -5
-10
0 500 1000 1500 2000 2500 0 500 1000 1500 2000 2500
Time [days; BJD - 2454833] Time [days; BJD - 2454833]
Fig. 3: From left to right and top to bottom: sinusoidal, dynamic, chaotic, and parabolic/sinusoidal classification of the TTVs. Note
that TTVs for KOI-0142.01 are given in days, rather than minutes.
metallicity and surface gravity, the values listed in the NASA ular, 30 equally spaced points were calculated and averaged to
Exoplanet Archive. Simultaneously to the transit model we fit- one data point. The modeling of all consecutive transits results in
ted a time-dependent second-order polynomial to account for a parameter distribution for the semi-major axis, the inclination,
out-of-transit variability. To determine reliable errors for the the orbital period and the planet-to-star radius ratio. We used
fitted parameters, we explored the parameter space by sam- their mean values and standard deviations to limit the ground-
pling from the posterior-probability distribution using a Markov- based data fitting (Section 4.2). All the orbital and physical pa-
chain Monte-Carlo (MCMC) approach. Our MCMC calcula- rameters computed for the 60 KOIs are summarized in the right
tions make extensive use of routines of PyAstronomy4, a col- part of Table 2. Errors are at the 1-σ level. It is worth to mention
lection of Python build-in functions that provide an interface that the transit parameters presented in the table provide us with
for fitting and sampling algorithms implemented in the PyMC an excellent transit template to be used to fit ground-based data.
(Patil et al. 2010) and SciPy (Jones et al. 2001) packages. We It is not our intention to improve any of the parameters by means
refer the reader to their detailed online documentation5. For of this simple analysis. A more detailed approach, such as photo-
the computation of the best-fit parameters we iterated 80 000 dynamical fitting might be required (see e.g. Barros et al. 2015),
times per consecutive transits, and discarded a conservative first specially with large-amplitude TTVs such as Kepler-9 (KOI-
20%. As starting values for the parameters we used the ones 0377.01/02, Holman et al. 2010; Ofir et al. 2014). As an illustra-
specified in the NASA Exoplanet Archive. To set reasonable tive example, Figure 4 shows how the transit parameters change
limits for MCMC’s uniform probability distributions, we chose as a function of time, evidencing their mutual correlations and
RP /RS ± 0.1, T0 ± TDur /3, and a considerable fraction of the or- the rate and amplitude at which they change. As expected, for
bital period, depending on the amplitude of Kepler TTVs. These the values in the figure the Pearson’s correlation coefficient be-
values are relative to the values determined by the Kepler team. tween the semi-major axis and the inclination is ra/R s ,i = 0.96,
The semi-major axis and the inclination are correlated through while these two reveal a strong anti-correlation with the planet-
the impact parameter, a/RS cos(i). Thus, rather than using uni- to-star radius ratio (rR p /R s ,i = -0.91, and rR p /R s ,a/R s = -0.93).
form distributions for these parameters we used Gaussian pri-
ors with the mean and the standard deviation equal to the val-
ues found in the NASA Exoplanet Archive and three times their 4.2. Primary transit fitting and detrending of ground-based
errors, respectively. To compute the transit parameters we ana- data
lyzed Kepler long cadence transit data. To minimize the impact Once DIP2 OL returns the photometric light curve and the asso-
of the sampling rate on the determination of the transit param- ciated detrending quantities, the computation of ground-based
eters (see e.g., Kipping 2010), during each instance of primary TTVs begins. First, we convert the time-axis, originally given
transit fitting we used a transit model calculated from a finer time in Julian dates, to Barycentric Julian dates using Eastman et al.
scale and then averaged on the Kepler timing points. In partic- (2010) web tool1 . To do so, we make use of the celestial coordi-
nates of the star, the geographic coordinates of the site, and the
4
http://www.hs.uni-hamburg.de/DE/Ins/Per/Czesla/ height above sea level. Throughout this work, our model com-
PyA/PyA/index.html
5 1
http://pymc-devs.github.io/pymc/ http://astroutils.astronomy.ohio-state.edu/time/utc2bjd.html
6C. von Essen et al. (2017): KOINet: study of exoplanet systems via TTVs
Table 2: Left: From left to right the KOI number, the right ascention, α, and the declination, δ, in degrees (J2000.0) and the Kepler
magnitude, K p . The values have been taken from the NASA Exoplanet Archive. Right: Best-fit orbital parameters obtained fitting
all available primary transits from quarter 1 to quarter 17 as described in this section. From left to right the semi-major axis in stellar
radii, a/RS , the inclination in degrees, i, the planet-to-star radius ratio, RP /RS , and the orbital period in days, Per. The last column,
O14-15, corresponds to the number of observations collected during 2014 and 2015.
KOI α (J2000) δ (J2000) Kp a/RS i RP /RS Per O14-15
Nr. (◦ ) (◦ ) (◦ ) (days)
0094.01 297.333069 41.891121 12.205 27.27 ± 0.03 89.997 ± 0.001 0.0691 ± 0.0001 22.34285 ± 0.00078 -
0094.03 297.333069 41.891121 12.205 50.5 ± 0.2 89.93 ± 0.01 0.0411 ± 0.0003 54.3198 ± 0.0018 -
0142.01 291.148071 40.669399 13.113 16.9 ± 0.9 87.4 ± 0.3 0.038 ± 0.001 10.947 ± 0.036 6
0250.01 284.940979 46.566540 15.473 32 ± 2 89.29 ± 0.07 0.051 ± 0.002 12.2827 ± 0.0044 2
0250.02 284.940979 46.566540 15.473 54 ± 6 89.3 ± 0.2 0.047 ± 0.005 17.2509 ± 0.0097 1
0315.01 297.271881 43.333309 12.968 59 ± 5 89.6 ± 0.2 0.029 ± 0.001 35.5812 ± 0.0087 4
0318.01 288.153992 44.068821 12.211 29.1 ± 0.2 89.9 ± 0.2 0.033 ± 0.003 38.5846 ± 0.0049 -
0345.01 286.524811 48.683601 13.340 45 ± 3 89.4 ± 0.2 0.0335 ± 0.0009 29.8851 ± 0.0031 -
0351.01 284.433502 49.305161 13.804 186.2 ± 0.1 89.970 ± 0.001 0.0852 ± 0.0001 331.616 ± 0.025 1
0351.02 284.433502 49.305161 13.804 141 ± 1 90.001 ± 0.001 0.0601 ± 0.0008 210.79 ± 0.41 1
0372.01 299.122437 41.866760 12.391 112 ± 1 89.98 ± 0.08 0.0816 ± 0.0009 125.6287 ± 0.0073 -
0377.01 285.573975 38.400902 13.803 33 ± 2 89.1 ± 0.2 0.078 ± 0.001 19.245 ± 0.023 12
0377.02 285.573975 38.400902 13.803 55 ± 6 89.3 ± 0.2 0.076 ± 0.003 38.95 ± 0.11 2
0410.01 292.248016 40.696049 14.454 33 ± 6 89.0 ± 0.9 0.065 ± 0.007 7.2165 ± 0.0018 6
0448.02 297.070160 40.868790 14.902 45 ± 10 88.9 ± 0.6 0.05 ± 0.01 43.587 ± 0.022 9
0456.01 287.773560 42.869282 14.619 20.4 ± 0.7 88.35 ± 0.07 0.034 ± 0.001 13.699 ± 0.012 3
0464.01 293.747101 45.107220 14.361 75.0 ± 0.3 89.95 ± 0.01 0.0677 ± 0.0008 58.3619 ± 0.0023 -
0523.01 286.047119 45.053211 15.000 45 ± 5 88.9 ± 0.2 0.063 ± 0.003 49.4112 ± 0.0082 1
0525.01 300.907776 45.457870 14.539 20 ± 2 87.3 ± 0.3 0.05 ± 0.01 11.5300 ± 0.0093 4
0528.02 287.101105 46.896481 14.598 102 ± 9 89.6 ± 0.1 0.031 ± 0.002 96.676 ± 0.010 -
0620.01 296.479767 49.937679 14.669 62.7 ± 0.4 89.90 ± 0.02 0.074 ± 0.001 45.1552 ± 0.0028 1
0620.02 296.479767 49.937679 14.669 127.2 ± 0.6 89.98 ± 0.01 0.1017 ± 0.0009 130.1783 ± 0.0058 -
0638.01 295.559418 40.236271 13.595 36.1 ± 0.3 89.65 ± 0.06 0.032 ± 0.001 23.6415 ± 0.0069 2
0738.01 298.348328 47.491230 15.282 27 ± 4 88.79 ± 0.07 0.037 ± 0.003 10.338 ± 0.015 4
0738.02 298.348328 47.491230 15.282 24 ± 2 88.33 ± 0.05 0.034 ± 0.005 13.286 ± 0.019 -
0757.02 286.999481 48.375790 15.841 68 ± 2 89.73 ± 0.07 0.046 ± 0.003 41.196 ± 0.011 -
0759.01 285.718536 48.504849 15.082 37 ± 4 88.8 ± 0.3 0.044 ± 0.003 32.628 ± 0.017 3
0760.01 292.167053 48.727589 15.263 12.2 ± 0.4 86.0 ± 0.2 0.106 ± 0.003 4.9592 ± 0.0012 7
0806.01 285.283630 38.947281 15.403 124 ± 7 89.84 ± 0.09 0.099 ± 0.001 143.200 ± 0.059 3
0806.02 285.283630 38.947281 15.403 75 ± 3 89.9 ± 0.1 0.136 ± 0.003 60.3258 ± 0.0062 4
0829.03 290.461761 40.562462 15.386 37 ± 3 88.7 ± 0.1 0.033 ± 0.003 38.557 ± 0.024 -
0841.01 292.236755 41.085880 15.855 31.3 ± 0.9 89.21 ± 0.07 0.054 ± 0.004 15.334 ± 0.011 1
0841.02 292.236755 41.085880 15.855 39 ± 5 88.9 ± 0.3 0.08 ± 0.01 31.3304 ± 0.0077 3
0854.01 289.508484 41.812119 15.849 89 ± 5 89.75 ± 0.05 0.041 ± 0.002 56.052 ± 0.021 1
0869.02 291.638977 42.436321 15.599 57 ± 2 89.64 ± 0.08 0.037 ± 0.001 36.277 ± 0.027 1
0880.01 292.873383 42.966141 15.158 36 ± 5 88.7 ± 0.4 0.045 ± 0.006 26.4435 ± 0.0097 1
0880.02 292.873383 42.966141 15.158 51 ± 6 89.3 ± 0.2 0.061 ± 0.002 51.537 ± 0.021 7
0886.01 294.773926 43.056301 15.847 8.9 ± 0.4 83.7 ± 0.1 0.07 ± 0.02 8.009 ± 0.015 1
0902.01 287.852386 43.897991 15.754 85 ± 7 89.6 ± 0.1 0.089 ± 0.002 83.927 ± 0.016 7
0918.01 283.977509 44.811562 15.011 51.6 ± 0.2 89.93 ± 0.02 0.116 ± 0.002 39.6432 ± 0.0016 3
0935.01 294.023010 45.853081 15.237 30.9 ± 0.5 89.5 ± 0.1 0.042 ± 0.001 20.860 ± 0.011 1
0935.02 294.023010 45.853081 15.237 40 ± 3 89.0 ± 0.2 0.042 ± 0.002 42.6334 ± 0.0081 -
0935.03 294.023010 45.853081 15.237 52 ± 7 89.1 ± 0.3 0.034 ± 0.002 87.647 ± 0.019 -
0984.01 291.048798 36.839882 11.631 20.6 ± 0.7 88.8 ± 0.1 0.030 ± 0.001 4.2888 ± 0.0031 8
1199.01 293.743927 38.939281 14.887 72 ± 2 89.77 ± 0.08 0.030 ± 0.002 53.526 ± 0.021 -
1271.01 294.265503 44.794300 13.632 105 ± 3 89.64 ± 0.02 0.0693 ± 0.0005 161.98 ± 0.16 4
1353.01 297.465332 42.882839 13.956 112 ± 3 89.85 ± 0.07 0.105 ± 0.001 125.8648 ± 0.0029 7
1366.01 286.358063 42.406509 15.368 29 ± 1 88.9 ± 0.1 0.031 ± 0.003 19.256 ± 0.019 1
1366.02 286.358063 42.406509 15.368 47 ± 12 88.7 ± 0.5 0.05 ± 0.02 54.156 ± 0.021 -
1426.01 283.209167 48.777641 14.232 48.3 ± 0.6 89.67 ± 0.09 0.029 ± 0.001 38.868 ± 0.011 -
1426.02 283.209167 48.777641 14.232 93 ± 11 89.6 ± 0.2 0.065 ± 0.002 74.927 ± 0.011 1
1426.03 283.209167 48.777641 14.232 131 ± 8 89.56 ± 0.05 0.12 ± 0.03 150.025 ± 0.013 -
1429.01 292.351501 48.511082 15.531 114 ± 21 89.6 ± 0.2 0.051 ± 0.003 205.914 ± 0.021 -
1474.01 295.417877 51.184761 13.005 50.4 ± 0.7 88.69 ± 0.05 0.25 ± 0.03 69.721 ± 0.032 2
1573.01 296.846161 40.138611 14.373 59.4 ± 0.9 89.80 ± 0.06 0.045 ± 0.001 24.8093 ± 0.0058 6
1574.01 297.916870 46.965130 14.600 60 ± 10 89.3 ± 0.2 0.067 ± 0.003 114.7356 ± 0.0079 -
1574.02 297.916870 46.965130 14.600 48 ± 7 88.9 ± 0.2 0.036 ± 0.001 191.29 ± 0.17 -
1873.01 295.809296 40.008511 15.674 63.9 ± 0.4 89.90 ± 0.02 0.045 ± 0.002 71.3106 ± 0.0087 -
2672.01 296.132812 48.977402 11.921 80 ± 12 89.5 ± 0.2 0.051 ± 0.003 88.508 ± 0.013 -
2672.02 296.132812 48.977402 11.921 72.2 ± 0.3 89.92 ± 0.01 0.0303 ± 0.0009 42.9933 ± 0.0042 -
7C. von Essen et al. (2017): KOINet: study of exoplanet systems via TTVs
and χ2 is computed from the residuals, obtained by subtracting
83.96
to the synthetic data the best-fit model. For the BIC and Cash, Q
Orbital period [days]
is the number of data points per light curve. The full detrending
83.94
model, DM, has the following expression:
83.92
DM(t) = c0 + c1 · χ̂ + c2 · Ŝ+
83.90
N+1
X (4)
95 bgi · BGi + fci · FCi + dki · DKi + xi · Xi + yi · Yi
i=1
92.5
RP/RS [ppt]
90 Here, N+1 denotes the total number of target and reference stars,
87.5 Ŝ and χ̂ correspond to seeing and airmass, respectively. Xi and
85 Yi are the (x,y) centroid positions. FCi and DKi are the inte-
82.5 grated flat and dark counts in the chosen aperture, respectively,
and BGi correspond to the background counts. The coefficients
90
of the detrending model are c0 , c1 , c2 , and bgi , f ci , dki and xi , yi ,
Inclination [deg]
89.8 with i = 1, N+1. Using a linear combination of these compo-
nents simplifies the computation of the detrending coefficients
89.6
that accompany them by means of simple inversion techniques.
89.4 Rather than using the full detrending model to clean the data
89.2
from systematics and potentially over-fit the data, we evaluate
110
sub-models of it (this is, a linear combination of some of the de-
Semi-major axis [a/RS]
trending components). Typical detrending functions would have
100
the following expression:
90
80
70 DM0 = c0 ,
60 DM1 = c0 + c1 χ̂ ,
0 400 800 1200 1600
DM2 = c0 + c1 χ̂ + c2 Ŝ ,
Time [days; BJD - 2454833]
DM3 = c0 + c2 Ŝ ,
Fig. 4: Time-dependent change of the transit parameters of N+1
X
KOI-0902.01. From top to bottom the orbital period in days in DM4 = c0 + bgi · BGi ,
triangles, the planet-to-star radius ratio, RP /RS in diamonds, the i=1
orbital inclination in squares, and the semi-major axis in stellar N+1
X
radii, a/RS . Horizontal continuous and dashed lines show mean DM5 = c0 + c1 χ̂ + bgi · BGi ,
and standard deviations of the system parameters, respectively. i=1
Individual errors are given at 1-σ level. ···
N+1
X
DM14 = c0 + c1 χ̂ + c2 Ŝ + bgi · BGi + xi · Xi + yi · Yi .
i=1
prises a primary transit times a detrending component. Thus, ···
to compute TTVs we carry out a more refined detrending of (5)
the light curves rather than just a time-dependent polynomial.
For the detrending model we consider a linear combination of DIP2 OL considers a total of 56 sub-models, depending on the
seeing, airmass, (x,y) centroid positions of the target and of the availability of calibrations. Usually, the noise in the data is cor-
reference stars, integrated counts over the selected photometric related with airmass, (x,y) centroid positions and integrated flat
aperture and the (x,y) centroid positions of the master flat field counts, while the dependency with seeing strongly depends on
and the master dark frames, when available, and integrated sky the photometric quality and stability of the particular night.
counts for the selected sky ring (see e.g., Kundurthy et al. 2013; Therefore, these 56 sub-models are constructed solely from how
Becker et al. 2013, for a similar approach in the detrending strat- we think the systematics impact the data. Although all possible
egy). Due to the nature of the data the exact time at which the combinations should be tested, this is computationally expen-
mid-transits will occur are in principle unknown, or known but sive, specially considering that a differential light curve can be
with a given certainty. Some photometric observations could ac- constructed averaging 20-30 reference stars (i.e., N = 20-30).
tually have been taken outside the primary transit occurrence. As To determine the detrending sub-model best matching the
a consequence, we have to be extremely careful not to over-fit residual noise in the data, we first create an array of trial T0 ’s
our data. In order to choose a sufficiently large number of fitting around the predicted mid-transit time, covering the ±Tdur space
parameters we take into consideration the joint minimization and respecting the cadence of the observations. This takes care of
of four statistical indicators: the reduced-χ2 statistic, χ2red , the the uncertainty in the knowledge of the mid-transit times, since
Bayesian Information Criterion, BIC = χ2 + k ln(Q), the stan- typical errors in the predictions of transits with large TTVs can
dard deviation of the residual light curves enlarged by the num- increase up to 40-50 minutes, in some cases even more. For each
ber of fitting parameters, σres ×k, and the Cash statistic (Cash one of these trial T0 ’s and each one of the sub-models we com-
PQ
1979), Cash = 2 i=1 Mi − Di ∗ ln(Mi ), being M the model and pute the previously mentioned four statistics. In principle, if a
D the data. For the BIC, k is the number of fitting parameters, given trial T0 is close to the true mid-transit time, then around
8C. von Essen et al. (2017): KOINet: study of exoplanet systems via TTVs
this T0 all the sub-models should minimize the four statistics. 6000 140
To illustrate this, Figure 5, top, shows how the BIC changes as
Number of detrending components
a function of the trial T0 , for all the possible sub-models (28 in 120
5000
this case, since dark frames were unavailable). For this example,
we analyzed the transit photometry of KOI-0760.01 taken with 100
the 2.5 m Nordic Optical Telescope. Color-coded are the number 4000
BIC value
of detrending components. Figure 5, bottom, shows the depen- 80
dency of the BIC with the sub-models (i.e., detrending models, 3000
DM). The numbers on the abscissa are in concordance with the 60
indices in Eq. 5. Color-coded are the trial T0 ’s. For this data set, 2000
the BIC minimizes at DM2 . As a consequence, the data do not 40
correlate with the integrated flat counts, nor the centroid posi- 1000 20
tion. This is actually what we expect, since the Nordic Optical
Telescope has an outstanding guiding system that can keep stars 0 0
within the same pixels for hours. -2 -1.5 -1 -0.5 0 0.5 1 1.5 2
Then, we make use of the minimization of the time-averaged Hours from predicted mid-transit time
statistics (that is, the statistics averaged within each one of the
T0 ’s) to determine the starting value of the mid-transit time that 6000 2
will be used in our posterior transit fitting (see Figure 6). This is a
more robust approach than simply computing the absolute min- 1.5
Trial mid-transit time [hours]
5000
imum value of the statistics, since these could be produced by 1
chance. Finally, with this mid-transit time fixed we re-compute 4000
the transit model and re-iterate over all the detrending models to 0.5
choose the one that minimizes the averaged statistics. BIC value
3000 0
For the transit fitting instance we use a quadratic limb dark-
ening law with quadratic limb darkening values computed as de- -0.5
scribed in von Essen et al. (2013), for the filter band matching 2000
the one used during the observations and for the stellar fun- -1
damental parameters closely matching the ones of the KOIs. 1000
-1.5
Rather than considering the orbital period, the inclination, the
semi-major axis and the planet-to-star radius ratio as fixed pa- 0 -2
rameters to the values given by the NASA Exoplanet Archive or 0 5 10 15 20 25
the values derived in Table 2, we use a Gaussian probability dis- Number of Detrending Model (DM)
tribution which mean and standard deviation equals the values
obtained in Section 4.1, and we fit all of them simultaneously to Fig. 5: Top: BIC values as a function of trial mid-transit times.
the detrending model and the mid-transit time. The inclination, Color-coded are the number of detrending components for each
semi-major axis and planet-to-star radius ratio are fitted only if one of the detrending models (sub-models). Bottom: BIC val-
the light curves show complete transit coverage. If not, we con- ues as a function of the detrending model, DM. Numbers are
sider them as fixed to the values reported in Table 2, and we fit in agreement with the labels on Eq. 5. Color-coded are the trial
only the mid-transit time. At each MCMC step the transit pa- T0 ’s.
rameters change. Therefore, for each iteration we compute the
detrending coefficients with the previously mentioned inversion
technique. To fit KOINet’s ground-based data we produce 5×106 6
repetitions of the MCMC chains, we discard the first 20%, and BIC
we compute the mean and standard deviation (1-σ) of the poste- 5
σresx k
rior distributions of the parameters as best-fit values and uncer- 2
χ red
Statistics
tainties, respectively. To check for the convergence of the chains, 4
we divide the remaining 80% in four, and we compute mean and Cash
standard deviations of the priors within each 20%. We consider 3
that the chains converged if all the values are consistent within 1-
σ errors. Finally, we visually inspect the posterior distributions 2
and their correlations.
1
To provide reliable error bars on the timing measurements -2 -1.5 -1 -0.5 0 0.5 1 1.5 2
we evaluate to what extent our photometric data are affected
Hours from predicted mid-transit time
by correlated noise. To this end, following Carter & Winn
(2009) we compute residual light curves by dividing our pho-
tometric data by the best-fit transit and detrending models. Fig. 6: The four statistics used to assess the number of detrend-
From the residuals, we compute the β factors as specified in ing components and the starting mid-transit time, obtained an-
von Essen et al. (2013). Here, we divide each residual light curve alyzing KOI-0760 data. Their values have been normalized and
into M bins of N averaged data points. If the data are free of scaled to allow for visual comparison. The thick dashed black
correlated noise, then the noise within the residual light curves line corresponds to the time-averaged BIC statistics, as shown in
should follow the expectation of independent random numbers: the top panel of Figure 5.
σ̂N = σ1 N −1/2 [M/(M − 1)]1/2 . (6)
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